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I know nothing about differential equations

  1. Aug 2, 2012 #1
    Hello

    I know nothing about differential equations, can you explain them to me in a few simple words?

    And what's so special so we study them seperately?

    Thanks
     
  2. jcsd
  3. Aug 2, 2012 #2
  4. Aug 2, 2012 #3
    Been there, done that

    I understand what an equation is.
    I bet an equation with variable(s), is a function (maybe formula too?).

    I cant get "differential" part.
    Is y=a*x+b a differential one ?
     
  5. Aug 2, 2012 #4
    A differential equation is basically an equation that has one or more derivatives in it, such as dy/dx + x = 3y.

    One reason we study them is because some equations arise where some function and its rate of change (its derivative) are related in the equation.
     
  6. Aug 2, 2012 #5
    I just think of a differential equation as a vector field or a field of arrows in space, telling you where to go.

    You just go where the arrows point and that's one way you can try to solve them.

    The glorified name for doing this approximately is "Euler's method", which isn't actually that practical in a lot of cases. But it gives you a good idea of what's going on.
     
  7. Aug 2, 2012 #6
    To expand on what homeomorphic said I'll give an example of a simple differential equation

    [itex]\frac{df}{dx}(x) = 1[/itex]

    This means that for every x the rate of change of f with respect to x is 1, this is also the same as saying the 'slope' of the graph of f(x) is always 1. The solution to this equation is f(x) = x.

    [itex]\frac{df}{dx}(x) = x[/itex]

    This means that the slope of the graph of f(x) is always equal to x, at x=1 the slope of f is 1, at x=10 the slope is 10. The solution to this one is [itex]f(x) = 2x^2[/itex] which is a little harder to see without knowing any calculus.

    The differential part just refers to the 'd' part in the equations, df is sometimes called the differential of f.
     
  8. Aug 2, 2012 #7
    Do you know what a function is? Let's stay in 1D for a moment. A function there is just some expression, formula, or rule that takes an input number (or a variable representing such a number) and spits out another number.

    Do you know what the derivative of a function is? It gives an idea of how fast the function's output changes with respect to changes in its input.

    Differential equations is the study of equations involving at least one derivative of a function, and usually based on just this information, we're trying to reconstruct what the original function might be or otherwise study the properties of such a function.
     
  9. Aug 3, 2012 #8
    [itex]f(x) = x + C[/itex]
    [itex]f(x) = \frac{1}{2}x^2 + C[/itex]
     
  10. Aug 3, 2012 #9
    whoops

    I purposly didn't add the C in though, as not to cause any undue confusion.
     
  11. Aug 3, 2012 #10
    Actually, the easiest way to get a feeling for differential equations is the following simplest differential equation:

    [itex] x'(t) = x(t) [/itex]

    Which function [itex] x(t) [/itex] fulfulls the above equation?
     
  12. Aug 3, 2012 #11
    I am not familiar with that notation
     
  13. Aug 3, 2012 #12
    Derivative is the rate of change of a variable?

    Eg if a variable changes like 2, 4, 6, 8, ... the rate of its change is 2?

    Also, equation with two or more variables inside is a function? (Ι bet there cannot be only one variable inside an equation and all the other parts of the equation being constants)
     
  14. Aug 3, 2012 #13
    Wow okay!

    Differentiate the following terms: [itex] x(t)=sin(t), x(t)=cos(t), x(t)=exp(t), x(t) = t, x(t) = t^2[/itex]
     
  15. Aug 3, 2012 #14
    Derivative is the rate of change of a function.

    For example, one might have a function [itex]f(x) = x^2[/itex]. This means the function [itex]f[/itex] takes whatever it is given and spits out the square. Hence, [itex]f(1) = 1[/itex], [itex]f(1.5) = 2.25[/itex], and [itex]f(\pi/2) = \pi^2/4[/itex].

    You can have linear functions. Let's say [itex]g(x) = 2x[/itex]. This defines a function so that [itex]g(1) = 2[/itex], [itex]g(2) = 4[/itex], [itex]g(3) = 6[/itex], and so on, but usually, unless one explicitly says so, it's perfectly fine to feed the function g a non-integer argument. In this case, [itex]g(1.5) = 3[/itex], for instance.

    To be honest, if you're unfamiliar with functions, I would study those for a while first before trying to understand derivatives and differential equations.
     
  16. Aug 3, 2012 #15
    Don’t confuse people x or t or what???
     
  17. Aug 3, 2012 #16
    Think about it like driving along the road, and recording how far you are from where you started. So at t = 0, when you first start recording, you're 0 miles from... somewhere. At another time, say t = 10 (call it minutes), you're now 5 miles from that initial spot.

    At any instant, you can record some distance you've traveled and plot it. If you're always going away from the start point (so your distance is always increasing), you have your distance -- we'll call it "d" -- as a function of time, t. Or, d = f(t).

    At the end, if you divided your total distance by the total time, you'd have your average velocity. Say for example you went 100 miles in 120 minutes (2 hours). That means you averaged 50 mph. In real life you're not going to go the same speed all the time, sometimes maybe 20 mph (hope I'm not stuck behind you), sometimes 75 mph, etc.

    So if you took the distance you'd traveled in 2 minutes (instead of the whole time), say between 2 and 4 minutes, you'd have your average velocity over that timespan. If you make that shorter and shorter (so your timespan of interest approaches zero), you are now finding the derivative (the instantaneous rate of change -- which in this case if velocity) of your distance function.

    Does that make sense?
     
  18. Aug 3, 2012 #17
    I think introducing a variable called "d" in a discussion about differentials may cause even more confusion! :wink:
     
  19. Aug 3, 2012 #18
    Don’t make things to difficult. It is just a simple equation. There are of course two ways to understand Des (from the math point also from the application point of view (but this is difficult) that’s what previous writer try to explain) .

    So, what is the solution of
    [itex] 3\cdot b = 6 [/itex] Every kid from play school knows the answer what b is.

    It is the same with Des. What is the solution of
    [itex] x’(t) = x(t)[/itex] .... ??? [itex] x(t) = ? [/itex]

    Is it maybe [itex] sin(t) [/itex] or [itex] exp(t) [/itex] or [itex] t^2 [/itex] ???

    I know that we had math students after 3 years who could not answer this question.
    They failed btw. Best answer was [itex] x(t) = \frac{1}{2} x(t) [/itex] which is definitly wrong. Why?

    After understanding this simple equation you could start to understand what differential equations
    are actually about. But first get familiar with the notation!
     
    Last edited: Aug 3, 2012
  20. Aug 3, 2012 #19
    The problem is that we have people unfamliar with what [itex]x'(t)[/itex] means (it denotes the first derivative of the function [itex]x(t)[/itex], and is also denoted [itex]\frac{dx}{dt}[/itex], among other ways).
     
  21. Aug 3, 2012 #20
    I think we're hinging on the equivalent of defining a word using the word in question (maddening!)

    Questions such as

    Makes it seem that the misunderstanding goes a bit deeper than "find a function x(t) that satisfies x'(t) = x(t)"...

    Yeah, the concepts you drew a comparison between are similar, but probably still a bit too top-level at this point.
     
  22. Aug 3, 2012 #21
    okay
    [itex] x'(t) = \frac{dx}{dt} = \frac{\partial x}{\partial t} = \frac{d}{dt} x(t) = \frac{\partial}{\partial t} x(t) = \dot{x}(t) [/itex].

    Aummarizing it is the derivation of a function [itex] x [/itex] after [itex] t [/itex]
     
  23. Aug 3, 2012 #22
    But how could you explain people what Des are. Starting with Law of mass action? Then you
    have to problem that everybody applies it but nobody really understand it what he is doing.
     
  24. Aug 3, 2012 #23
    That's not to say you can't define them ever. But start where they're more likely to be familiar with the subject. I'm not saying what I wrote is best, either.

    But if he isn't familiar with x'(t) notation, or what a function is, then giving all the different notations of a derivative might be confusing. I know it would have confused me. Then you get the "what's dx/dy? or that funny looking d (partials!) mean?" Ya dig? He or she will start getting caught up in the details rather than the concepts.

    I just remember when I first started pondering DEs. And I got the formal definitions at first and they didn't help (they did, but it was harder to wrap my brain around it. I still haven't succeeded). I had to move from something specific and then say "Oh, that's like 'this' and 'that' and... Oh I see now." and then move onto the rest.
     
  25. Aug 3, 2012 #24
    Although there are already very good responses, I would like to add my take.
    We solve simple algebraic equations like this
    x^2+2x + 1 = 5
    It means, find a number which if squared and added to its double and added 1 is equal to 5.
    Differential Equations are just like that but they apply to changing quantity like distance traveled, Voltage in AC mains etc. The special feature of such things, unlike the variable x we discussed earlier is that they are changing with time and have rate of change.
    For eg. rate of change of Voltage V is 5V per second, which will be written as
    dv/dt = 5;

    If we form equations with such quantity incorporating the rate of change in the equations, then the puzzle formed is called differential equation.
    Eg. dv/dt + 2*v = 4;
    The solution cant give something like V = 5 volts because the voltage is changing constantly.
    So, the soultion should actually give V at various times, like
    V = 5 volts at t = 0.1 seconds
    V = 6 Volts at t = 0.2 seconds
    V = ...
    etc.
    So, instead of providing solution table, we state the solution mathematically, like
    V = 3*sin(t) + 2*t
    Now put the required time 't' in right side and obtain V at that time.
    We now call, V is a function of 't'.
    The task of differential equations is to solve for the function V.
     
  26. Aug 4, 2012 #25
    Okay! I was not aware of that. Sorry but then forget it!

    I mean the same would be: I like to play poker but I don't know what cards or money are!
     
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